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Chapter 16

ECON 6002 Econometrics Memorial University of Newfoundland. Qualitative and Limited Dependent Variable Models. Chapter 16. Adapted from Vera Tabakova’s notes . Chapter 16: Qualitative and Limited Dependent Variable Models. 16.1 Models with Binary Dependent Variables

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Chapter 16

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  1. ECON 6002 Econometrics Memorial University of Newfoundland Qualitative and Limited Dependent Variable Models Chapter 16 Adapted from Vera Tabakova’s notes

  2. Chapter 16: Qualitative and Limited Dependent Variable Models • 16.1 Models with Binary Dependent Variables • 16.2 The Logit Model for Binary Choice • 16.3 Multinomial Logit • 16.4 Conditional Logit • 16.5 Ordered Choice Models • 16.6 Models for Count Data • 16.7 Limited Dependent Variables Principles of Econometrics, 3rd Edition

  3. 16.1 Models with Binary Dependent Variables • Examples: • An economic model explaining why some individuals take a second, or third, job and engage in “moonlighting.” • An economic model of why the federal government awards development grants to some large cities and not others. • An economic model explaining why someone is in the labour force or not Principles of Econometrics, 3rd Edition

  4. 16.1 Models with Binary Dependent Variables • An economic model explaining why some loan applications are accepted and others not at a large metropolitan bank. • An economic model explaining why some individuals vote “yes” for increased spending in a school board election and others vote “no.” • An economic model explaining why some female college students decide to study engineering and others do not. Principles of Econometrics, 3rd Edition

  5. 16.1 Models with Binary Dependent Variables As long as these exhaust the possible (mutually exclusive) options If the probability that an individual drives to work is p, then It follows that the probability that a person uses public transportation is . Principles of Econometrics, 3rd Edition

  6. 16.1.1 The Linear Probability Model Principles of Econometrics, 3rd Edition

  7. 16.1.1 The Linear Probability Model One problem with the linear probability model is that the error term is heteroskedastic; the variance of the error term e varies from one observation to another. Principles of Econometrics, 3rd Edition

  8. 16.1.1 The Linear Probability Model Using generalized least squares, the estimated variance is: So the problem of heteroskedasticity is not insurmountable… Principles of Econometrics, 3rd Edition

  9. 16.1.1 The Linear Probability Model Principles of Econometrics, 3rd Edition

  10. 16.1.1 The Linear Probability Model Principles of Econometrics, 3rd Edition Slide16-10 Problems: We can easily obtain values of that are less than 0 or greater than 1 Some of the estimated variances in (16.6) may be negative, so the WLS would not work Of course, the errors are not distributed normally R2 is usually very poor and a questionable guide for goodness of fit

  11. 16.1.2 The Probit Model Figure 16.1 (a) Standard normal cumulative distribution function (b) Standard normal probability density function Principles of Econometrics, 3rd Edition

  12. 16.1.2 The Probit Model Principles of Econometrics, 3rd Edition

  13. 16.1.3 Interpretation of the Probit Model where and is the standard normal probability density function evaluated at Note that this is clearly a nonlinear model: the marginal effect varies depending on where you measure it Principles of Econometrics, 3rd Edition

  14. 16.1.3 Interpretation of the Probit Model Equation (16.11) has the following implications: • Since is a probability density function its value is always positive. Consequently the sign of dp/dx is determined by the sign of 2. In the transportation problem we expect 2 to be positive so that dp/dx > 0; as x increases we expect p to increase. Principles of Econometrics, 3rd Edition

  15. 16.1.3 Interpretation of the Probit Model • As x changes the value of the function Φ(β1 + β2x) changes. The standard normal probability density function reaches its maximum when z = 0, or when β1 + β2x = 0. In this case p = Φ(0) = .5 and an individual is equally likely to choose car or bus transportation. The slope of the probit function p = Φ(z) is at its maximum when z = 0, the borderline case. Principles of Econometrics, 3rd Edition

  16. 16.1.3 Interpretation of the Probit Model • On the other hand, if β1 + β2x is large, say near 3, then the probability that the individual chooses to drive is very large and close to 1. In this case a change in x will have relatively little effect since Φ(β1 + β2x) will be nearly 0. The same is true if β1 + β2x is a large negative value, say near 3. These results are consistent with the notion that if an individual is “set” in their ways, with p near 0 or 1, the effect of a small change in commuting time will be negligible. Principles of Econometrics, 3rd Edition

  17. 16.1.3 Interpretation of the Probit Model Predicting the probability that an individual chooses the alternative y = 1: Although you have to be careful with this Interpretation! Principles of Econometrics, 3rd Edition

  18. 16.1.4 Maximum Likelihood Estimation of the Probit Model Suppose that y1 = 1, y2 = 1 and y3 = 0. Suppose that the values of x, in minutes,are x1 = 15, x2 = 20 and x3 = 5. Principles of Econometrics, 3rd Edition

  19. 16.1.4 Maximum Likelihood Estimation of the Probit Model In large samples the maximum likelihood estimator is normally distributed, consistent and best, in the sense that no competing estimator has smaller variance. Principles of Econometrics, 3rd Edition

  20. 16.1.5 An Example Principles of Econometrics, 3rd Edition

  21. 16.1.5 An Example Measured at DTIME = 20 Principles of Econometrics, 3rd Edition

  22. 16.1.5 An Example If it takes someone 30 minutes longer to take public transportation than to drive to work, the estimated probability that auto transportation will be selected is Since this estimated probability is 0.798, which is greater than 0.5, we may want to “predict” that when public transportation takes 30 minutes longer than driving to work, the individual will choose to drive. But again use this cautiously! Principles of Econometrics, 3rd Edition

  23. 16.1.5 An Example Principles of Econometrics, 3rd Edition Slide16-23 In STATA: Use transport.dta

  24. 16.1.5 An Example Linear fit???  Slide16-24

  25. 16.1.5 An Example Principles of Econometrics, 3rd Edition Slide16-25

  26. 16.1.5 An Example Slide16-26

  27. 16.1.5 An Example * marginal effects mfx mfx,at (dtime=30) * direct calculation nlcom (normalden(_b[_cons]+_b[dtime]*30)*_b[dtime] ) and nlcom (normal(_b[_cons]+_b[dtime]*30) ) Slide16-27

  28. 16.2 The Logit Model for Binary Choice Principles of Econometrics, 3rd Edition

  29. 16.2 The Logit Model for Binary Choice so Principles of Econometrics, 3rd Edition

  30. 16.2 The Logit Model for Binary Choice so So the “logit”, the log-odds, is actually a fully linear function of X  Principles of Econometrics, 3rd Edition Slide16-30

  31. As Probability goes from 0 to 1, logit goes from –infinite to + infinite • The logit is linear, but the probability is not • The explanatory variables are individual specific, but do not change across alternatives • The slope coefficient tells us by how much the log-odds changes with a unit change in the variable

  32. Slide16-32 This model can be in principle estimated with WLS (due to the heteroskedasticity in the error term) if we have grouped data (glogit in STATA, while blogit will run ML logit on grouped data) Otherwise we use MLE on individual data

  33. Goodness of fit McFadden’s pseudo R2 Count R2 (% of correct predictions) Etc. Measures of goodness of fit are of secondary importance What counts is the sign of the regression coefficients and their statistical and practical significance

  34. Goodness of fit Using MLE A large sample method => estimated errors are asymptotic => we use Z test statistics (based on the normal distribution), instead of t statistics A likelihood ratio test (with a test statistic distributed as chi-square with df= number of regressors) is equivalent to the F test

  35. Goodness of fit: example See http://www.soziologie.uni-halle.de/langer/logitreg/books/long/stbfitstat.pdf

  36. Goodness of fit: example But be very careful with these measures! So in STATA The “ones” do not Really have to be Actual ones, just Non-zeros

  37. More diagnostics To compute the deviance of the residuals: predict “newname”, deviance The deviance for a logit model is like the RSS in OLS. The smaller the deviance the better the fit. And (Logit only) to combine with information about leverage: predict “newnamedelta”, ddeviance (A recommended cut-off value for the ddeviance is 4)

  38. More diagnostics

  39. Probit versus Logit

  40. Probit versus Logit A matter of taste nowadays, since we all have good computers The underlying distributions share the mean of zero but have different variances: Logit And normal 1 So estimated slope coefficients differ by a factor of about 1.8 ( ) . Logit ones are bigger

  41. More on Probit versus Logit Watch out for “perfect predictions” Luckily STATA will flag them for you  and drop the culprit observations

  42. More on Probit versus Logit Learn about the test (Wald tests based on chi-2)and lrtest commands (LR tests), so you can test hypotheses as we did with t-tests and F tests in OLS They are asymptotically equivalent but can differ in small samples

  43. More on Probit versus Logit Learn about the many extra STATA capabilities that will make your postestimation life much easier Long and Freese’s book is a great resource

  44. For example  Principles of Econometrics, 3rd Edition Slide16-44

  45. For example  Principles of Econometrics, 3rd Edition Slide16-45

  46. More on Probit versus Logit Go through a couple of examples available online with your own STATA session connected to the internet. Examples: http://www.ats.ucla.edu/stat/stata/dae/probit.htm http://www.ats.ucla.edu/stat/stata/dae/logit.htm http://www.ats.ucla.edu/stat/stata/output/old/lognoframe.htm http://www.ats.ucla.edu/stat/stata/output/stata_logistic.htm

  47. Keywords • binary choice models • censored data • conditional logit • count data models • feasible generalized least squares • Heckit • identification problem • independence of irrelevant alternatives (IIA) • index models • individual and alternative specific variables • individual specific variables • latent variables • likelihood function • limited dependent variables • linear probability model • logistic random variable • logit • log-likelihood function • marginal effect • maximum likelihood estimation • multinomial choice models • multinomial logit • odds ratio • ordered choice models • ordered probit • ordinal variables • Poisson random variable • Poisson regression model • probit • selection bias • tobit model • truncated data Principles of Econometrics, 3rd Edition

  48. References Long, S. and J. Freese for all topics (available on Google!)

  49. Next Multinomial Logit Conditional Logit

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